Solving Nonlinear PDEs with Sparse Radial Basis Function Networks
This work addresses challenges in PDE solving for computational science by offering an adaptive, learning-inspired method, though it appears incremental as it blends existing strengths from RBF, PINNs, and GP approaches.
The paper tackles solving nonlinear PDEs by proposing a sparse radial basis function network framework that uses sparsity-promoting regularization to prevent over-parameterization, with numerical experiments showing effectiveness and advantages over Gaussian process approaches in some cases.
We propose a novel framework for solving nonlinear PDEs using sparse radial basis function (RBF) networks. Sparsity-promoting regularization is employed to prevent over-parameterization and reduce redundant features. This work is motivated by longstanding challenges in traditional RBF collocation methods, along with the limitations of physics-informed neural networks (PINNs) and Gaussian process (GP) approaches, aiming to blend their respective strengths in a unified framework. The theoretical foundation of our approach lies in the function space of Reproducing Kernel Banach Spaces (RKBS) induced by one-hidden-layer neural networks of possibly infinite width. We prove a representer theorem showing that the sparse optimization problem in the RKBS admits a finite solution and establishes error bounds that offer a foundation for generalizing classical numerical analysis. The algorithmic framework is based on a three-phase algorithm to maintain computational efficiency through adaptive feature selection, second-order optimization, and pruning of inactive neurons. Numerical experiments demonstrate the effectiveness of our method and highlight cases where it offers notable advantages over GP approaches. This work opens new directions for adaptive PDE solvers grounded in rigorous analysis with efficient, learning-inspired implementation.